Weak Metrics on Euclidean Domains
نویسندگان
چکیده
A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce a weak metric, called the Apollonian weak metric, on any subset of a Euclidean space which is either bounded or whose boundary is unbounded. We relate this weak metric to some familiar metrics such as the Poincaré metric, the Klein-Hilbert metric, the Funk metric and the part metric which all play important roles in classical and in recent work on geometric function theory. This paper will appear in the JP Jour. Geometry & Topology AMS Mathematics Subject Classification: 30F45, 51M15, 51K99.
منابع مشابه
ar X iv : m at h / 06 09 23 6 v 1 [ m at h . M G ] 8 S ep 2 00 6 WEAK METRICS ON EUCLIDEAN DOMAINS
A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. In the present paper we introduced a weak metric, called the Apollonian weak metric, on any subset of a Euclidean space which is either bounded or whose boundary is unbounded. We then relate this weak metric to some familiar metrics such as the Poincaré metric, the Klein-Hil...
متن کاملCompact lsospectral Sets of Surfaces
In this paper we study sets of surfaces which are isospectral with respect to the Laplace-Beltrami operator. More specifically, for closed surfaces (compact, no boundary) we consider a fixed surface and the family of metrics on that surface having a given Laplace spectrum, whereas for surfaces with boundary we confine our study to the class of simply connected planar domains all having the same...
متن کاملHermitian-einstein Metrics for Vector Bundles on Complete Kähler Manifolds
In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermi...
متن کاملThe Funk and Hilbert geometries for spaces of constant curvature
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...
متن کاملAssessment of the Log-Euclidean Metric Performance in Diffusion Tensor Image Segmentation
Introduction: Appropriate definition of the distance measure between diffusion tensors has a deep impact on Diffusion Tensor Image (DTI) segmentation results. The geodesic metric is the best distance measure since it yields high-quality segmentation results. However, the important problem with the geodesic metric is a high computational cost of the algorithms based on it. The main goal of this ...
متن کامل